A direct N-body integrator for modelling the chaotic, tidal dynamics of multibody extrasolar systems: TIDYMESS

Tidal dissipation plays an important role in the dynamical evolution of moons, planets, stars, and compact remnants. The interesting complexity originates from the interplay between the internal structure and external tidal forcing. Recent and upcoming observing missions of exoplanets and stars in the galaxy help to provide constraints on the physics of tidal dissipation. It is timely to develop new N-body codes, which allow for experimentation with various tidal models and numerical implementations. We present the open-source N-body code TIDYMESS, which stands for ‘TIdal DYnamics of Multibody ExtraSolar Systems’. This code implements a Creep deformation law for the bodies, parametrized by their fluid Love numbers and fluid relaxation times. Due to tidal and centrifugal deformations, we approximate the general shape of a body to be an ellipsoid. We calculate the associated gravitational field to quadruple order, from which we derive the gravitational accelerations and torques. The equations of motion for the orbits, spins and deformations are integrated directly using a fourth-order integration method based on a symplectic composition. We implement a novel integration method for the deformations, which allows for a time-step solely dependent on the orbits, and not on the spin periods or fluid relaxation times. This feature greatly speeds up the calculations, while also improving the consistency when comparing different tidal regimes. We demonstrate the capabilities and performance of TIDYMESS, particularly in the niche regime of parameter space where orbits are chaotic and tides become non-linear

Gargantuan chaotic gravitational three-body systems and their irreversibility to the Planck length

Chaos is present in most stellar dynamical systems and manifests itself through the exponential growth of small perturbations. Exponential divergence drives time irreversibility and increases the entropy in the system. A numerical consequence is that integrations of the N-body problem unavoidably magnify truncation and rounding errors to macroscopic scales. Hitherto, a quantitative relation between chaos in stellar dynamical systems and the level of irreversibility remained undetermined. In this work, we study chaotic three-body systems in free fall initially using the accurate and precise N-body code Brutus, which goes beyond standard double-precision arithmetic. We demonstrate that the fraction of irreversible solutions decreases as a power law with numerical accuracy. This can be derived from the distribution of amplification factors of small initial perturbations. Applying this result to systems consisting of three massive black holes with zero total angular momentum, we conclude that up to 5 per cent of such triples would require an accuracy of smaller than the Planck length in order to produce a time-reversible solution, thus rendering them fundamentally unpredictable

Relativistic Pythagorean three-body problem

We study the influence of relativity on the chaotic properties and dynamical outcomes of an unstable triple system; the Pythagorean three-body problem. To this end, we extend the brutus N-body code to include post-Newtonian pairwise terms up to 2.5 order, and the first order Taylor expansion to the Einstein-Infeld-Hoffmann equations of motion. The degree to which our system is relativistic depends on the scaling of the total mass (the unit size was 1 parsec). Using the brutus method of convergence, we test for time-reversibility in the conservative regime, and demonstrate that we are able to obtain definitive solutions to the relativistic three-body problem. It is also confirmed that the minimal required numerical accuracy for a successful time-reversibility test correlates with the amplification factor of an initial perturbation, as was found previously for the Newtonian case. When we take into account dissipative effects through gravitational wave emission, we find that the duration of the resonance, and the amount of exponential growth of small perturbations depend on the mass scaling. For a unit mass , the system behavior is indistinguishable from Newton’s equations of motion, and the resonance always ends in a binary and one escaping body. For a mass scaling up to , relativity gradually becomes more prominent, but the majority of the systems still dissolve in a single body and an isolated binary. The first mergers start to appear for a mass of , and between and all systems end prematurely in a merger. These mergers are preceded by a gravitational wave driven in-spiral. For a mass scaling , all systems result in a gravitational wave merger upon the first close encounter. Relativistic three-body encounters thus provide an efficient pathway for resolving the final parsec problem. The onset of mergers at the characteristic mass scale of potentially leaves an imprint in the mass function of supermassive black holes

Reversible time-step adaptation for the integration of few-body systems

The time-step criterion plays a crucial role in direct N-body codes. If not chosen carefully, it will cause a secular drift in the energy error. Shared, adaptive time-step criteria commonly adopt the minimum pairwise time-step, which suffers from discontinuities in the time evolution of the time-step. This has a large impact on the functioning of time-step symmetrization algorithms. We provide new demonstrations of previous findings that a smooth and weighted average over all pairwise time-steps in the N-body system, improves the level of energy conservation. Furthermore, we compare the performance of 27 different time-step criteria, by considering three methods for weighting time-steps and nine symmetrization methods. We present performance tests for strongly chaotic few-body systems, including unstable triples, giant planets in a resonant chain, and the current Solar System. We find that the harmonic symmetrization methods (methods A3 and B3 in our notation) are the most robust, in the sense that the symmetrized time-step remains close to the time-step function. Furthermore, based on our Solar System experiment, we find that our new weighting method based on direct pair-wise averaging (method W2 in our notation), is slightly preferred over the other methods

Stellar triples on the edge: Comprehensive overview of the evolution of destabilised triples leading to stellar and binary exotica

Context. Hierarchical triple stars are ideal laboratories for studying the interplay between orbital dynamics and stellar evolution. Both mass loss from stellar winds and strong gravitational perturbations between the inner and outer orbit cooperate to destabilise triple systems. Aims. Our current understanding of the evolution of unstable triple systems is mainly built upon results from extensive binary-single scattering experiments. However, destabilised hierarchical triples cover a different region of phase space. Therefore, we aim to construct a comprehensive overview of the evolutionary pathways of destabilised triple-star systems. Methods. Starting from generic initial conditions, we evolved an extensive set of hierarchical triples using the code TRES, combining secular dynamics and stellar evolution. We detected those triples that destabilise due to stellar winds and/or gravitational perturbations. Their evolution was continued with a direct N-body integrator coupled to stellar evolution. Results. The majority of triples (54–69%) preserve their hierarchy throughout their evolution, which is in contradiction with the commonly adopted picture that unstable triples always experience a chaotic, democratic resonant interaction. The duration of the unstable phase was found to be longer than expected (103 − 4 crossing times, reaching up to millions), so that long-term stellar evolution effects cannot be neglected. The most probable outcome is dissolution of the triple into a single star and binary (42–45%). This occurs through the commonly known democratic channel, during which the initial hierarchy is lost and the lightest body usually escapes, but also through a hierarchical channel, during which the tertiary is ejected in a slingshot, independent of its mass. Collisions are common (13–24% of destabilised triples), and they mostly involve the two original inner binary components still on the main sequence (77–94%). This contradicts the idea that collisions with a giant during democratic encounters dominate (only 5–12%). Together with collisions in stable triples, we find that triple evolution is the dominant mechanism for stellar collisions in the Milky Way. Lastly, our simulations produce runaway and walk-away stars with speeds up to several tens of km/s, with a maximum of a few 100 km s−1. We suggest that destabilised triples can explain – or at least alleviate the tension behind – the origin of the observed (massive) runaway stars. Conclusions. A promising indicator for distinguishing triples that will follow the democratic or hierarchical route, is the relative inclination between the inner and outer orbits. Its influence can be summed up in two rules of thumb: (1) prograde triples tend to evolve towards hierarchical collisions and ejections, and (2) retrograde triples tend to evolve towards democratic encounters and a loss of initial hierarchy, unless the system is compact, which experience collision preferentially. The trends found in this work complement those found previously from binary-single scattering experiments, and together they will help to generalise and improve our understanding on the evolution of unstable triple systems of various origins

On the Jacobi capture origin of binaries with applications to the Earth-Moon system and black holes in galactic nuclei

Close encounters between two bodies in a disc often result in a single orbital deflection. However, within their Jacobi volumes, where the gravitational forces between the two bodies and the central body become competitive, temporary captures with multiple close encounters become possible outcomes: a Jacobi capture. We perform 3-body simulations in order to characterise the dynamics of Jacobi captures in the plane. We find that the phase space structure resembles a Cantor-like set with a fractal dimension of about 0.4. The lifetime distribution decreases exponentially, while the distribution of the closest separation follows a power law with index 0.5. In our first application, we consider the Jacobi capture of the Moon. We demonstrate that both tidal captures and giant impacts are possible outcomes. The impact speed is well approximated by a parabolic encounter, while the impact angles follow that of a uniform beam on a circular target. Jacobi captures at larger heliocentric distances are more likely to result in tidal captures. In our second application, we find that Jacobi captures with gravitational wave dissipation can result in the formation of binary black holes in galactic nuclei. The eccentricity distribution is approximately super-thermal and includes both prograde and retrograde orientations. We conclude that dissipative Jacobi captures form an efficient channel for binary formation, which motivates further research into establishing the universality of Jacobi captures across multiple astrophysical scales.Comment: Accepted by MNRAS. 19 pages, 16 figure

Formation of supermassive black hole seeds in nuclear star clusters via gas accretion and runaway collisions

More than 200 supermassive black holes (SMBHs) of masses ≳109M⊙≳109M⊙ have been discovered at z ≳ 6. One promising pathway for the formation of SMBHs is through the collapse of supermassive stars (SMSs) with masses ∼103−105M⊙∼103−105M⊙ into seed black holes which could grow upto few times 109M⊙109M⊙ SMBHs observed at z ∼ 7. In this paper, we explore how SMSs with masses ∼103−105M⊙∼103−105M⊙ could be formed via gas accretion and runaway stellar collisions in high-redshift, metal-poor nuclear star clusters (NSCs) using idealized N-body simulations. We explore physically motivated accretion scenarios, e.g. Bondi–Hoyle–Lyttleton accretion and Eddington accretion, as well as simplified scenarios such as constant accretions. While gas is present, the accretion time-scale remains considerably shorter than the time-scale for collisions with the most massive object (MMO). However, overall the time-scale for collisions between any two stars in the cluster can become comparable or shorter than the accretion time-scale, hence collisions still play a crucial role in determining the final mass of the SMSs. We find that the problem is highly sensitive to the initial conditions and our assumed recipe for the accretion, due to the highly chaotic nature of the problem. The key variables that determine the mass growth mechanism are the mass of the MMO and the gas reservoir that is available for the accretion. Depending on different conditions, SMSs of masses ∼103−105M⊙∼103−105M⊙ can form for all three accretion scenarios considered in this work

Formation of SMBH seeds in Population III star clusters through collisions: the importance of mass loss

Runaway collisions in dense clusters may lead to the formation of supermassive black hole (SMBH) seeds, and this process can be further enhanced by accretion, as recent models of SMBH seed formation in Population III star clusters have shown. This may explain the presence of SMBHs already at high redshift, z > 6. However, in this context, mass loss during collisions was not considered and could play an important role for the formation of the SMBH seed. Here, we study the effect of mass loss, due to collisions of protostars, in the formation and evolution of a massive object in a dense primordial cluster. We consider both constant mass-loss fractions as well as analytic models based on the stellar structure of the collision components. Our calculations indicate that mass loss can significantly affect the final mass of the possible SMBH seed. Considering a constant mass loss of 5 per cent for every collision, we can lose between 60–80 per cent of the total mass that is obtained if mass loss were not considered. Using instead analytical prescriptions for mass loss, the mass of the final object is reduced by 15–40 per cent, depending on the accretion model for the cluster we study. Altogether, we obtain masses of the order of 104M⊙104M⊙⁠, which are still massive enough to be SMBH seeds

Chaos in self-gravitating many-body systems Lyapunov time dependence of N and the influence of general relativity

In self-gravitating N-body systems, small perturbations introduced at the start, or infinitesimal errors that are produced by the numerical integrator or are due to limited precision in the computer, grow exponentially with time. For Newton's gravity, we confirm earlier results that for relatively homogeneous systems, this rate of growth per crossing time increases with N up to N 7sim; 30, but that for larger systems, the growth rate has a weaker scaling with N. For concentrated systems, however, the rate of exponential growth continues to scale with N. In relativistic self-gravitating systems, the rate of growth is almost independent of N. This effect, however, is only noticeable when the system's mean velocity approaches the speed of light to within three orders of magnitude. The chaotic behavior of systems with more than a dozen bodies for the usually adopted approximation of only solving the pairwise interactions in the Einstein-Infeld-Hoffmann equation of motion is qualitatively different than when the interaction terms (or cross terms) are taken into account. This result provides a strong motivation for follow-up studies on the microscopic effect of general relativity on orbital chaos, and on the influence of higher-order cross-terms in the Taylor-series expansion of the Einstein-Infeld-Hoffmann equations of motion

Formation of massive seed black holes via collisions and accretion

Models aiming to explain the formation of massive black hole seeds, and in particular the direct collapse scenario, face substantial difficulties. These are rooted in rather ad hoc and fine-tuned initial conditions, such as the simultaneous requirements of extremely low metallicities and strong radiation backgrounds. Here, we explore a modification of such scenarios where a massive primordial star cluster is initially produced. Subsequent stellar collisions give rise to the formation of massive (104−105 M⊙) objects. Our calculations demonstrate that the interplay among stellar dynamics, gas accretion, and protostellar evolution is particularly relevant. Gas accretion on to the protostars enhances their radii, resulting in an enhanced collisional cross-section. We show that the fraction of collisions can increase from 0.1 to 1 per cent of the initial population to about 10 per cent when compared to gas-free models or models of protostellar clusters in the local Universe. We conclude that very massive objects can form in spite of initial fragmentation, making the first massive protostellar clusters viable candidate birth places for observed supermassive black holes